VARIATION AND OSCILLATION FOR SINGULAR INTEGRALS WITH ODD KERNEL ON LIPSCHITZ GRAPHS

VARIATION AND OSCILLATION FOR SINGULAR INTEGRALS WITH ODD KERNEL

ON LIPSCHITZ GRAPHS

ALBERT MAS AND XAVIER TOLSA

Abstract. We prove that, for ρ > 2, the ρ-variation and oscillation for the smooth trun- cations of the Cauchy transform on Lipschitz graphs is bounded in L^pfor 1 < p < ∞. The analogous result holds for the n-dimensional Riesz transform on n-dimensional Lipschitz graphs, as well as for other singular integral operators with odd kernel. In particular, our results strengthen the classical theorem on the L²boundedness of the Cauchy transform on Lipschitz graphs by Coifman, McIntosh, and Meyer.

1. Introduction

The ρ-variation and oscillation for martingales and some families of operators have been studied in many recent papers on probability, ergodic theory, and harmonic analysis (see [L´e], [Bo], [JKRW], [CJRW1], and [JSW], for example). The purpose of this paper is to establish some new results concerning the ρ-variation and oscillation for families of singular integral operators defined on Lipschitz graphs. In particular, our results include the L^p boundedness of the ρ-variation and the oscillation for the smooth truncations of the Cauchy transform and the n-dimensional Riesz transform on Lipschitz graphs, for 1 < p <∞ and ρ > 2.

Given a Borel measure µ in R^d, one defines the n-dimensional Riesz transform of a function f ∈ L¹(µ) by R^µf (x) = lim_ǫց0R^µ_ǫf (x) (whenever the limit exists), where

R^µ_ǫf (x) = Z

|x−y|>ǫ

x− y

|x − y|ⁿ⁺¹f (y) dµ(y), x∈ R^d.

When d = 2 (i.e., µ is a Borel measure in C), one defines the Cauchy transform of f ∈ L¹(µ) by C^µf (x) = lim_ǫց0Cǫ^µf (x) (whenever the limit exists), where

C_ǫ^µf (x) = Z

|x−y|>ǫ

f (y)

x− ydµ(y), x∈ C.

To avoid the problem of existence of the preceding limits, it is useful to consider the maximal operators R_∗^µf (x) = sup_ǫ>0|R^µǫf (x)| and C_∗^µf (x) = sup_ǫ>0|Cǫ^µf (x)|.

The Cauchy and Riesz transforms are two very important examples of singular integral operators with a Calder´on-Zygmund kernel. The kernels K : R^d\ {0} → R that we consider in this paper satisfy

(1.1) |K(x)| ≤ C

|x|ⁿ, |∂xⁱK(x)| ≤ C

|x|ⁿ⁺¹ and |∂xⁱ∂_xjK(x)| ≤ C

|x|ⁿ⁺²,

Date: March, 2011.

1991 Mathematics Subject Classification. Primary 42B20, 42B25.

Key words and phrases. ρ-variation, oscillation, Calder´on-Zygmund singular integrals.

Both authors are partially supported by grants 2009SGR-000420 (Generalitat de Catalunya) and MTM2010-16232 (Spain). Albert Mas is also supported by grant AP2006-02416 (FPU program, Spain).

1

for all 1≤ i, j ≤ d and x = (x¹, . . . , x^d)∈ R^d\{0}, where 0 < n < d is some integer and C > 0 is some constant; and moreover K(−x) = −K(x) for all x 6= 0 (i.e. K is odd). Notice that the n-dimensional Riesz transform corresponds to the vector kernel (x¹, . . . , x^d)/|x|ⁿ⁺¹, and the Cauchy transform to (x¹,−x²)/|x|² (so, we may consider K to be any scalar component of these vector kernels).

Given an odd kernel K satisfying (1.1) and a finite Borel measure µ in R^d, for each ǫ > 0, we consider the ǫ-truncated operator

T_ǫµ(x) = Z

|x−y|>ǫ

K(x− y) dµ(y), x∈ R^d,

and then we set T µ(x) = lim_ǫց0Tǫµ(x) whenever the limit makes sense, and T_∗µ(x) = sup_ǫ>0|Tǫµ(x)|. Finally, given f ∈ L¹(µ), we define Tǫ^µf (x) := T_ǫ(f µ)(x), T^µf (x) :=

T (f µ)(x) and T_∗^µf (x) := T_∗(f µ)(x). Thus, for a suitable choice of K, the operator T^µ coincides with the Cauchy or Riesz transforms.

Besides the operator T_ǫ defined above, one can consider another ǫ-truncated variant that we proceed to define. First we need some additional notation. Given x = (x¹, . . . , x^d)∈ R^d, we use the notation ex := (x¹, . . . , xⁿ) ∈ Rⁿ. Let ϕ_R : [0,∞) → [0, ∞) be a non decreasing C² function such that χ_[3^√_n,∞) ≤ ϕR≤ χ[2.1√

n,∞) (the numbers 3√

n and 2.1√

n are chosen just for definiteness and they are not important). Given ǫ > 0 and x∈ R^d, we denote

ϕǫ(x) := ϕ_R(|ex|/ǫ) and ϕ := {ϕǫ}ǫ>0. Given K as above, x∈ R^d, 0 < ǫ, and a finite Borel measure µ, we set

(Kϕ_ǫ∗ µ)(x) :=

Z

ϕ_ǫ(x− y)K(x − y) dµ(y).

We also denote (Kϕ∗ µ)(x) := {(Kϕǫ ∗ µ)(x)}ǫ>0. Finally, given f ∈ L¹(µ), we define Tϕ^µǫf (x) := (Kϕ_ǫ ∗ (fµ))(x), Tϕ^µf (x) := lim_ǫ→0Tϕ^µǫf (x) (whenever the limit makes sense), T_ϕ^µ∗f (x) := sup_ǫ>0|Tϕ^µǫf (x)|, and Tϕ^µf (x) :={Tϕ^µǫf (x)}ǫ>0.

Let I be a subset of R (in this paper, we will always have I = (0, ∞) or I = Z), and let F := {Fǫ}ǫ∈I be a family of functions defined on R^d. Given ρ > 0, the ρ-variation of F at x∈ R^dis defined by

Vρ(F)(x) := sup

{ǫ^m}

 X

m∈Z

|Fǫm+1(x)− Fǫm(x)|^ρ

1/ρ

,

where the pointwise supremum is taken over all decreasing sequences{ǫm}m∈Z ⊂ I. Fix a decreasing sequence{rm}m∈Z⊂ I. The oscillation of F at x ∈ R^d is defined by

O(F)(x) := sup

{ǫ^m},{δ^m}

 X

m∈Z

|Fǫm(x)− Fδm(x)|²

1/2

,

where the pointwise supremum is taken over all sequences{ǫm}m∈Z ⊂ I and {δm}m∈Z⊂ I such that r_m+1 ≤ ǫm≤ δm ≤ rm for all m∈ Z.

In this paper we are interested in studying the ρ-variation and oscillation for the family Tϕ^µf . That is, we will deal with

(Vρ◦ Tϕ^µ)f (x) :=Vρ(Tϕ^µf )(x) =Vρ(Kϕ∗ (fµ))(x) and (O ◦ Tϕ^µ)f (x) :=O(Tϕ^µf )(x) =O(Kϕ ∗ (fµ))(x),

for a Borel measure µ and f ∈ L¹(µ). Although it is not clear from the definitions, these operators are µ-measurable (see [CJRW1], [JSW]).

Given E ⊂ R^d, we denote by Hⁿ_E the n-dimensional Hausdorff measure restricted to E.

Let Γ := {x ∈ R^d : x = (ex, A(ex))} be the graph of a Lipschitz function A : Rⁿ → R^d−n with Lipschitz constant Lip(A). Let H¹(Hⁿ_Γ) and BM O(Hⁿ_Γ) be the (atomic) Hardy space and the space of functions with bounded mean oscillation, respectively, with respect to the measureHⁿ_Γ. The following is our main result.

Theorem 1.1. Let ρ > 2, let K be a kernel satisfying (1.1), and set µ :=HⁿΓ. The operators Vρ◦ Tϕ^µ andO ◦ Tϕ^µ are bounded

• in L^p(µ) for 1 < p <∞,

• from H¹(µ) to L¹(µ),

• from L¹(µ) to L^1,∞(µ), and

• from L^∞(µ) to BM O(µ).

In all the cases above, the norm of O ◦ Tϕ^µ is bounded independently of the sequence that defines O.

Let us recall that the L²(H¹Γ) boundedness of the Cauchy transform on Lipschitz graphs Γ⊂ C with slope small enough was proved by A. P. Calder´on in his celebrated paper [Ca].

The L²boundedness on Lipschitz graphs in full generality was proved later on by R. Coifman, A. McIntosh, and Y. Meyer [CMM].

Consider the Cauchy kernel K(z) = 1/z (z ∈ C), and set µ := H¹_Γ, so Cǫ^µ = Tǫ^µ. By standard Calder´on-Zygmund theory (namely, Cotlar’s inequality), the L²(µ) boundedness of the Cauchy transform C^µ is equivalent to the L²(µ) boundedness of the maximal operator C_∗^µ. Let M^µ denote the Hardy-Littlewood maximal operator with respect to the measure µ. It is easy to check that, for f ∈ L¹(µ) with compact support, there exists some constant C₀ > 0 such that

C_ǫ^µf (x)≤ Tϕ^µǫf (x) + C0M^µf (x)≤ (Vρ◦ Tϕ^µ)f (x) + C0M^µf (x)

for all ǫ > 0, thus (Vρ◦ Tϕ^µ) + C₀M^µ controls the maximal operator C_∗^µ and, in this sense, Theorem 1.1 (together with the known L^p(µ) boundedness of M^µ) strengthens the results of [Ca] and [CMM]. Analogous conclusions hold for the n-dimensional Riesz transform and the maximal operator R^µ_∗.

The operator Vρ◦ Tϕ^µ is also related to an important open problem posed by G. David and S. Semmes which actually is our main motivation to prove Theorem 1.1. We need some definitions to state it.

Recall that a measure µ is said to be n-dimensional Ahlfors-David regular, or simply AD regular, if there exists some constant C such that C⁻¹rⁿ ≤ µ(B(x, r)) ≤ Crⁿ for all x∈ suppµ and 0 < r ≤ diam(suppµ). It is not difficult to see that such a measure µ must be of the form µ = hHⁿsuppµ, where h is some positive function bounded above and away from zero. A Borel set E ⊂ R^d is called AD regular if the measureHⁿ_E is AD regular. One says that µ is uniformly n-rectifiable, or simply uniformly rectifiable, if there exist θ, M > 0 so that, for each x∈ suppµ and R > 0, there is a Lipschitz mapping g from the n-dimensional ball Bⁿ(0, R) ⊂ Rⁿ into R^d such that Lip(g) ≤ M and µ B(x, R) ∩ g(Bⁿ(0, R))

≥ θRⁿ, where Lip(g) stands for the Lipschitz constant of g. In the language of [DS2], this means that suppµ has big pieces of Lipschitz images of Rⁿ. A Borel set E ⊂ R^d is called uniformly n- rectifiable ifHⁿ_E is n-uniformly rectifiable. Of course, the n-dimensional graph of a Lipschitz function is uniformly n-rectifiable.

David and Semmes asked the following question, which is still open (see [Pa, Chapter 7]):

Problem 1.2. Is it true that an n-dimensional AD regular measure µ is n-uniformly recti- fiable if and only if R^µ_∗ is bounded in L²(µ)?

It is proved in [DS1] that if µ is uniformly rectifiable, then R^µ_∗ is bounded in L²(µ).

However, the converse implication has been proved only in the case n = 1 and d = 2, by P. Mattila, M. Melnikov and J. Verdera [MMV], using the notion of curvature of measures (which seems to be useful only in this case).

Set R^µ := {R^µǫ}ǫ>0. By combining some techniques from [DS2] and [To], in our forth- coming paper [MT] we show that the L² boundedness ofVρ◦ R^µimplies that µ is uniformly n-rectifiable. Moreover, we also prove thatVρ◦ R^µ is bounded in L²(µ) for all AD regular uniformly n-rectifiable measures µ. So we obtain the following theorem, which might be considered as a first approach to a possible solution of Problem 1.2:

Theorem. Let ρ > 2. An n-dimensional AD regular measure µ is uniformly n-rectifiable if and only if Vρ◦ R^µ is a bounded operator in L²(µ).

An essential ingredient for the proof of this result is Theorem 1.1 above. The arguments and techniques used to derive the L²boundedness ofVρ◦R^µon uniformly rectifiable measures from the L² boundedness ofVρ◦ R^µϕ on Lipschitz graphs are quite delicate (R^µϕ is defined as R^µ but using the family ϕ for the truncations). In particular, they involve the corona type decomposition introduced in [DS1]. For this reason, the proof of the preceding theorem is out of the scope of this paper and will appear in [MT].

Concerning the background on the ρ-variation and oscillation, a fundamental result is L´epingle’s inequality [L´e], from which the L^p boundedness of the ρ-variation and oscillation for martingales follows, for ρ > 2 and 1 < p <∞ (see Theorem 2.4 below for more details).

From this result on martingales, one deduces that the ρ-variation and oscillation are also bounded in L^pfor the averaging operators (also called differentiation operators, see [JKRW]):

(1.2) D_ǫf (x) = 1

|B(x, ǫ)|

Z

B(x,ǫ)

f (y) dy, x∈ R.

As far as we know, the first work dealing with the ρ-variation and oscillation for singular in- tegral operators is the one of J. Campbell, R. L. Jones, K. Reinhold and M. Wierdl [CJRW1], where the L^p and weak L¹ boundedness of the ρ-variation (for ρ > 2) and oscillation for the Hilbert transform was proved. Recall that, for f ∈ L^p(R) and x∈ R,

H_ǫf (x) = 1 π

Z

|x−y|>ǫ

1

x− yf (y) dy,

and then the Hilbert transform of f is defined by Hf (x) = lim_ǫ→0H_ǫf (x), whenever the limit exists. Later on, there appeared other papers showing the L^p boundedness of the ρ- variation and oscillation for singular integrals in R^d ([CJRW2]), with weights ([GT]), or for other operators such as the spherical averaging operator or singular integral operators on parabolas ([JSW]). Finally, we remark that, very recently, the case of the Carleson operator has been considered too ([LT], [OSTTW]).

Notice that the Hilbert transform is one of the simplest examples where Theorem 1.1 applies (one sets Γ = R, i.e. A≡ 0), and so one obtains a new proof of the L^p boundedness of the ρ-variation and oscillation for the Hilbert transform. In the original proof in [CJRW1], a key ingredient was the following classical identity, which follows via the Fourier transform:

(1.3) Qǫ = Pǫ∗ H,

where P_ǫ is the Poisson kernel and Q_ǫ is the conjugated Poisson kernel. Using this identity and the close relationship between the operators Qǫ and Hǫ, Campbell et al. derived the L^p boundedness of the ρ-variation and oscillation for the Hilbert transform from the one of

the family{Dǫ(Hf )}ǫ>0, where D_ǫ is the averaging operator in (1.2) (notice that P_ǫ can be written as a convex combination of operators D_δ, δ > 0).

In most of the previous results concerning ρ-variation and oscillation of families of oper- ators from harmonic analysis, the Fourier transform is a fundamental tool. However, this is not useful in order to prove Theorem 1.1, since the graph Γ is not invariant under translations in general. Moreover, even for the Cauchy transform, there is no formula like (1.3), which relates the truncations of a singular integral operator with an averaging operator applied to a singular integral operator, when Γ is a general Lipschitz graph.

The main ingredients of our proof of Theorem 1.1 are the known results on the ρ-variation and oscillation for martingales (L´epingle’s inequality [L´e]) and a multiscale analysis which stems from the geometric proof of the L² boundedness of the Cauchy transform on Lipschitz graphs by P. W. Jones [Jn1] and his celebrated work [Jn2] on quantitative rectifiability in the plane, using the so called β coefficients. Some of the techniques in these papers were further developed in higher dimensions by David and Semmes [DS1] for Ahlfors-David regular sets. More recently, in [To] some coefficients denoted by α, in the spirit of the Jones’ β’s, were introduced, and they were shown to be useful for the study of the L^p-boundedness of Calder´on-Zygmund operators on Lipschitz graphs and on uniformly rectifiable sets (see the definition below Theorem 1.3). In our paper, the α and β coefficients play a fundamental role.

Let us remark that L´epingle’s inequality, which asserts the L^p boundedness of the ρ- variation of martingales, fails if one assumes ρ≤ 2 (see [Qi] and [JW], for example). More- over, this fact can be brought to the ρ-variation of averaging operators and singular integral operators, thus it is essential to assume ρ > 2 in Theorem 1.1. Analogous conclusions hold if one replaces the ℓ²-norm by and ℓ^ρ-norm with ρ < 2 in the definition ofO. See [CJRW1], or [AJS] for the case of martingales.

Concerning the direct applications of Theorem 1.1, it is easily seen that the L^p bound- edness of Vρ ◦ Tϕ^µ yields a new proof of the existence of the principal values Tϕ^µf (x) :=

lim_ǫ→0Tϕ^µǫf (x) for all f ∈ L^p(µ) and almost all x ∈ Γ, without using a dense class of func- tions in L^p(µ) as in the classical proof. Moreover, from Theorem 1.1 one also gets some information on the speed of convergence. In fact, a classical result derived from variational inequalities is the boundedness of the λ-jump operator N_λ◦ Tϕ^µ and the (a, b)-upcrossings operator N_a^b◦ Tϕ^µ. Given λ > 0, f ∈ L¹_loc(µ) and x ∈ R^d, one defines (N_λ◦ Tϕ^µ)f (x) as the supremmum of all integers N for which there exists 0 < ǫ₁ < δ₁ ≤ ǫ2 < δ₂ ≤ · · · ≤ ǫN < δ_N so that

|Tϕ^µ_ǫif (x)− Tϕ^µ_δif (x)| > λ

for each i = 1, . . . , N . Similarly, given a < b, one defines (N_a^b◦Tϕ^µ)f (x) to be the supremmum of all integers N for which there exists 0 < ǫ₁ < δ₁ ≤ ǫ2 < δ₂ ≤ · · · ≤ ǫN < δ_N so that T_ϕ^µ_ǫif (x) < a and T_ϕ^µ_δif (x) > b for each i = 1, . . . , N . Using Theorem 1.1 one obtains (by the same arguments as in [CJRW1, Theorem 1.3 and Corollary 7.1]) the following:

Theorem 1.3. Let ρ > 2, λ > 0, and let K, and µ be as in Theorem 1.1. For 1 < p <∞, there exist constants C₁ and C₂ depending on ρ, n, d, K, and Lip(A) (and on p for the case of C₁) such that

k (Nλ◦ Tϕ^µ)f
1/ρ

kL^p(µ) ≤ C₁

λ kfkL^p(µ) and µ({x ∈ Γ : (Nλ◦ Tϕ^µ)f (x) > m}) ≤ C₂

λm^1/ρ kfkL¹(µ).

Trivially, (N_a^b ◦ Tϕ^µ)f ≤ (N_b−a◦ Tϕ^µ)f , thus Theorem 1.3 also holds replacing λ by b− a and N_λ by N_a^b. In [JSW] it is shown that the results of Theorem 1.3 still hold when ρ = 2 for the particular case of the Hilbert transform. In our paper we do not pursue this endpoint result.

2. Preliminaries

Throughout all the paper, n and d are two fixed integers such that 0 < n < d. Given a point x = (x¹, . . . , x^d) ∈ R^d, we use the notation ex := (x¹, . . . , xⁿ) ∈ Rⁿ. Given a function f : R^m→ R, we denote by ∇f its gradient (when it makes sense), and by ∇²f the matrix of second derivatives of f . If f depends on different points x₁, x₂, . . .∈ R^m, then∇xif denotes the gradient of f with respect to the x_i variable, and analogously for ∇²xif .

For two sets F₁, F₂ ⊂ R^d, we denote by dist_H(F₁, F₂) the Hausdorff distance between F₁ and F₂. We denote by Lⁿ the Lebesgue measure on Rⁿ, and for the sake of simplicity, we setk · kp :=k · kL^p(Lⁿ) for 1≤ p ≤ ∞, and dy := dLⁿ(y) for y∈ Rⁿ.

In the paper, when we refer to the angle between two affine n-planes in R^d, we mean the angle between the n-dimensional subspaces associated to the n-planes. As usual, the letter

‘C’ stands for some constant which may change its value at different occurrences, and which quite often only depends on n and d. The notation A . B (A & B) means that there is some fixed constant C such that A≤ CB (A ≥ CB), with C as above. Also, A ≈ B is equivalent to A . B . A.

2.1. More about the family ϕ. Given x∈ R^d, 0 < ǫ ≤ δ, and a finite Borel measure µ, we set ϕ^δ_ǫ(x) := ϕ_ǫ(x)− ϕδ(x) and we define

(Kϕ^δ_ǫ ∗ µ)(x) :=

Z

ϕ^δ_ǫ(x− y)K(x − y) dµ(y), thus (Kϕ^δ_ǫ∗ µ)(x) = (Kϕǫ∗ µ)(x) − (Kϕδ∗ µ)(x).

For m ∈ N, x ∈ R^m, and R ≥ r > 0, we denote by B^m(x, r) the closed ball of R^m with center x and radius r, and by A^m(x, r, R) the closed annulus of R^m centered at x with inner radius r and outer radius R. We also use the notation B(x, r) and A(x, r, R) when there is no possible confusion about m.

Each function ϕ^δ_ǫ is non negative, and suppϕ^δ_ǫ ⊂ Aⁿ(0, 2.1ǫ√ n, 3δ√

n) × R^d−n ⊂ R^d. Moreover, P

j∈Zϕ₂²−⁻j−1^j (x) = 1 for ex 6= 0, and there are at most two terms that do not vanish in the previous sum for a given x∈ R^d.

2.2. The α and β coefficients. Special dyadic lattice. Given m∈ N, λ > 0, and a cube Q⊂ R^m (i.e. Q := [0, b)^m+ a with a∈ R^m and b > 0), ℓ(Q) denotes the side length of Q, z_Q denotes the center of Q and λQ denotes the cube with center z_Q and side length λℓ(Q).

Throughout the paper, we will only use cubes with sides parallel to the axes.

Let µ be a locally finite Borel measure on R^d. Given 1≤ p < ∞ and a cube Q ⊂ R^d, one sets (see [DS2])

(2.1) β_p,µ(Q) = inf

L

 1

ℓ(Q)ⁿ Z

2Q

dist(y, L) ℓ(Q)

p

dµ(y)

1/p

,

where the infimum is taken over all n-planes L in R^d. For p =∞ one replaces the L^p norm by the supremum norm:

(2.2) β_∞,µ(Q) = inf

L

 sup

y∈suppµ∩2Q

dist(y, L) ℓ(Q)

 ,

where the infimum is taken over all n-planes L in R^dagain. These coefficients were introduced by P. W. Jones in [Jn1] for p =∞ and by G. David and S. Semmes in [DS1] for 1 ≤ p < ∞.

Let F ⊂ R^d be the closure of an open set. Given two finite Borel measures σ, ν on R^d, one sets

(2.3) distF(σ, ν) := supn R

f dσ−R

f dν : Lip(f) ≤ 1, suppf ⊂ F o

.

It is easy to check that this is a distance in the space of finite Borel measures σ such that suppσ⊂ F and σ(∂F ) = 0. Moreover, it turns out that this distance is a variant of the well known Wasserstein distance W₁ from optimal transportation (see [Vi, Chapter 1]). See [Ma, Chapter 14] for other properties of distF.

Given a cube Q which intersects suppµ, consider the closed ball B_Q := B(z_Q, 6ℓ(Q)).

Then one defines (see [To])

(2.4) αⁿ_µ(Q) := 1

ℓ(Q)ⁿ⁺¹ inf

c≥0,Ldist_BQ(µ, cHⁿL),

where the infimum is taken over all constants c≥ 0 and all n-planes L in R^d. For convenience, if Q does not intersect suppµ, we set αⁿ_µ(Q) = 0. To simplify notation, sometimes we will write α_µ(Q) or α(Q) instead of αⁿ_µ(Q) (and analogously for the β’s).

The following result characterizes uniform rectifiability in terms of the α and β coefficients.

Theorem 2.1. Let µ be an n-dimensional AD regular measure on R^d, and consider any p∈ [1, 2]. Then, the following are equivalent:

(a) µ is uniformly n-rectifiable.

(b) For any cube R⊂ R^d,

(2.5) X

Q∈D_Rd(R)

β_p,µ(Q)²ℓ(Q)ⁿ≤ Cℓ(R)ⁿ

with C independent of R, where D_R^d(R) stands for the collection of cubes of R^d contained in R which are obtained by splitting R dyadically.

(c) There exists C > 0 such that, for any cube R⊂ R^d,

(2.6) X

Q∈D_Rd(R)

α_µ(Q)²ℓ(Q)ⁿ≤ Cℓ(R)ⁿ.

The equivalence (a)⇐⇒(b) in Theorem 2.1 was proved by G. David and S. Semmes in [DS1], and the equivalence (a)⇐⇒(c) was proved by X. Tolsa in [To].

In this paper we will use a slightly different definition of the α and β coefficients adapted to the n-uniformly rectifiable measure µ = fHΓⁿ, where Γ := {x ∈ R^d : x = (ex, A(ex))} is the n-dimensional graph of a given Lipschitz function A : Rⁿ → R^d−n and f ∈ L^∞(Hⁿ_Γ) satisfies f (x)≈ 1 for almost all x ∈ Γ. To this end, we need to introduce a special dyadic lattice of sets related to Γ. Given a cube eQ ⊂ Rⁿ (i.e. Q := [0, b)e ⁿ + a with a ∈ Rⁿ and b > 0), we define Q := eQ× R^d−n. This type of set will be called v-cube (“vertical”

cube). We denote by ℓ(Q) and ez_Q the side length and center of eQ, respectively, and given λ > 0 we set λQ := λ eQ× R^d−n. Let eD denote the standard dyadic lattice of Rⁿ, and set D := {Q : eQ ∈ eD}. It is easy to check that the v-cubes of D intersected with Γ provide a dyadic lattice associated to the graph Γ in the sense of [Da, Appendix 1]. Finally, for m∈ Z, setDm:={Q ∈ D : ℓ(Q) = 2^−m}.

Fix a constant C_Γ > 10√

n(1 + Lip(A)) (the precise value of C_Γwill not be relevant in the proofs given in the paper). Given 1≤ p ≤ ∞ and a v-cube Q ⊂ R^d, we define the coefficient

β_p,µ(Q) as in (2.1) and (2.2) but replacing 2Q by C_ΓQ. We also define α_µ(Q) as in (2.4) but taking B_Q := B(ez_Q, C_Γℓ(Q))× R^d−n ⊂ R^d. This new definition of the α and β coefficients (adapted to the graph Γ) is the one that we will use in the whole paper.

Remark 2.2. It is an exercise to check that, with this new definition of the α’s and β’s, inequalities (2.5) and (2.6) of Theorem 2.1 still hold. Moreover, the following is an easy consequence of (2.5) and (2.6): Let Γ be an n-dimensional Lipschitz graph, f ∈ L^∞(HⁿΓ) such that f (x)≈ 1 for almost all x ∈ Γ, and µ = fH_Γⁿ. Let 1≤ p ≤ 2. Given C1, C2, C3 ≥ 1, there exists a constant C₄ > 0 such that, for any R∈ D,

X

Q∈D: Q⊂C1R

β_p,µ(C₂Q)²+ α_µ(C₃Q)²

µ(Q)≤ C4µ(R), and the dependence of C4 with respect to Γ is only on Lip(A).

Remark 2.3. It is shown in [To, Lemma 3.2], that β1,µ(Q) . αµ(Q) for all Q ∈ D. Given Q∈ D, let LQbe a minimizing n-plane for α_µ(Q). In general, β_∞,µ(Q) can not be controlled by β_1,µ(Q), so given x∈ suppµ ∩ CΓQ, we can not control dist(x, L_Q) by means of α_µ(Q).

But it is shown in [To, Lemma 5.2] that dist(x, L_Q) . P

R∈D: x∈R⊂Qα_µ(R)ℓ(R), and in particular, if P ∈ D is such that P ⊂ Q and x ∈ suppµ ∩ CΓP , and L_P denotes a minimizing n-plane for α_µ(P ), one has (see [To, Remark 5.3])

dist(x, L_Q) . dist(x, L_P) + X

R∈D: P ⊂R⊂Q

α_µ(R)ℓ(R).

(2.7)

2.3. Martingales. First of all, let us recall a particular case of L´epingle’s inequality (see [JSW], or [L´e] and [JKRW, Theorem 6.4] for martingales in a probability space):

Theorem 2.4. Let (X, Σ, λ) be a σ-finite measure space and ρ > 2. Then, there exist constants C₁, C₂ > 0 such that, for every martingale G := {Gm}_m∈Z∈ L²(λ),

kVρ(G)kL²(λ) ≤ C1kGkL²(λ) and kO(G)kL²(λ) ≤ C2kGkL²(λ),

where kGkL²(λ) := sup_m∈ZkGmkL²(λ). The constants C₁ and C₂ do not depend on the mea- sure λ, and C₂ neither depends on the fixed sequence that defines O.

To prove Theorem 1.1, we need to introduce a particular martingale, and to review some known results.

Lemma 2.5. Fix a cube eP ⊂ Rⁿ (not necessarily dyadic) and a Lipschitz graph Γ := {x ∈ R^d : x = (ex, A(ex))} such that suppA ⊂ eP . Consider the measure µ := fHⁿ_Γ, where f (x) = 1 for all ex∈ eP^c and C₀⁻¹ ≤ f(x) ≤ C0 for all ex ∈ eP , for some fixed constant C₀ > 0. Also set P := eP× R^d−n. Then, the following hold:

T_∗µ∈ L¹loc(µ), T_∗(χ_Eµ)∈ L¹loc(µ) for every compact set E⊂ R^d, and (2.8)

kT µkL²(µ) .µ(P )^1/2. (2.9)

Remark 2.6. To avoid the problem of non-integrability near infinity, for this type of measures µ we redefine T_ǫµ(x) := lim_{M →∞}R

χ_{(ǫ,M )}(|x − y|)K(x − y) dµ(y), which exists because µ is flat outside a compact set and K is odd. All the results in this paper remain valid with this new definition and the adjustments that have to be done in the proofs are minimal.

In this paper, we will deal with other integrals which concern the kernel K and the measure µ near infinity. The non-integrability problem can be avoided in the same manner.

Proof of Lemma 2.5. It is known that the operator T_∗^µis bounded in L²(µ), because T_∗^µis the maximal operator associated to a Calder´on-Zygmund singular integral and µ is an uniformly rectifiable measure (see [DS1]). Thus, T_∗(χ_Eµ) = T_∗^µ(χ_E) ∈ L¹_loc(µ) for every compact set E⊂ R^d.

We are going to check that kT∗µkL²(µ) . µ(P )^1/2. This will imply that T_∗µ ∈ L¹_loc(µ) and, since T µ exists (because µ is uniformly rectifiable) and|T µ| ≤ T_∗µ, we will also obtain kT µkL²(µ) .µ(P )^1/2; so the lemma will be proved.

Using that T_∗^µ is bounded in L²(µ), we have

kT_∗µkL²(µ) ≤ kT_∗(χ_3Pµ)kL²(µ)+kT_∗(χ_{(3P )}^cµ)kL²(µ)

.µ(P )^1/2+kT∗(χ_{(3P )}^cµ)kL²(µ). (2.10)

Set L := Rⁿ× {0}^d−n ⊂ R^d; obviously χ_P^cµ =Hⁿ_L\P. Since L is an n-plane and K is odd, T_∗HⁿL(x) = 0 for all x∈ L. Thus,

kT∗Hⁿ_L\3PkL²(HⁿL)≤ kT∗HⁿLkL²(HⁿL)+kT∗Hⁿ_L∩3PkL²(HⁿL).µ(P )^1/2. (2.11)

Set zP := (ezP, 0, . . . , 0) ∈ L (recall that ezP denotes the center of eP ) and χǫ(x) :=

χ_(ǫ,∞)(|x|). It is obvious that R

χ_ǫ(z_P − y)K(zP − y) dHⁿ_L\3P(y) = 0 for all ǫ > 0. Thus, given x∈ suppµ ∩ P ,

|(Kχǫ∗ H_L\3Pⁿ )(x)| ≤ Z

χ_ǫ(x− y)|K(x − y) − K(zP − y)| dHⁿ_L\3P(y) +

Z

|χǫ(x− y) − χǫ(z_P − y)||K(zP − y)| dHⁿ_L\3P(y).

Since Γ is a Lipschitz graph, |x − zP| . ℓ(P ). So, the first term on right hand side of the previous inequality is easily bounded by an absolute constant independent of ǫ, by standard arguments. For the second term, notice that supp(χ_ǫ(x− ·) − χǫ(z_P − ·)) ∩ (L \ 3P ) = ∅ for all ǫ < ℓ(P ), and HLⁿ({y ∈ Rⁿ : χ_ǫ(x− y) − χǫ(z_P − y) 6= 0}) . ℓ(P )ǫⁿ⁻¹ for all ǫ≥ ℓ(P ).

Therefore, since|zP − y| ≈ ǫ for all y ∈ supp(χǫ(x− ·) − χǫ(z_P − ·)) ∩ (L \ 3P ), the second term can also be estimated by an absolute constant. Thus, we conclude T_∗Hⁿ_L\3P(x) = sup_ǫ>0|(Kχǫ∗ H_L\3Pⁿ )(x)| . 1 for all x ∈ suppµ ∩ P .

Using the previous observations and (2.11), we have

kT∗(χ_{(3P )}^cµ)k²L²(µ) =kT∗Hⁿ_L\3Pk²L²(χPµ)+kT∗Hⁿ_L\3Pk²L²(χP cµ)

≤ kT∗Hⁿ_L\3Pk²L²(χPµ)+kT∗Hⁿ_L\3Pk²L²(HⁿL).µ(P ),

which, combined with (2.10), giveskT∗µkL²(µ) .µ(P )^1/2, as desired.  We are ready to define the martingale. Let P and µ be as in Lemma 2.5. Given m ∈ Z and a∈ Rⁿ, we set

De_m^a := a + [0, 2^−m)ⁿ⊂ Rⁿ and D_m^a := eD_m^a × R^d−n⊂ R^d.

Set Dm^a := {Dm^a+2−mk ⊂ R^d : k ∈ Zⁿ} (notice that Dm^a coincides with Dm translated by a parameter a ∈ Rⁿ and, for a fixed a, S

m∈ZDm^a is a translation of the standard dyadic lattice). Notice that µ(D_m^a)≈ 2^−mn for all m∈ Z, a ∈ Rⁿ. For D∈ Dm^a and x∈ D, we set

E_Dµ(x) := 1 µ(D)

Z

D

Z

D^c

K(z− y) dµ(y) dµ(z) (take into account Remark 2.6 for the meaning ofR

D^cK(z− y) dµ(y)). Finally, for x ∈ R^d, we define the martingale E_m^aµ(x) :=P

D∈Dm^a χ_D(x)E_Dµ(x), m∈ Z.

Let us make some comments to understand better the nature of E_m^aµ. First of all notice that, since µ(∂D) = 0, for any D∈ Dm^a and µ-almost all z∈ D we have

(2.12)

Z

D^c

K(z− y) dµ(y) = lim

ǫ→0

Z

D^c

χ_ǫ(z− y)K(z − y) dµ(y), and for any ǫ > 0, we have

(2.13)

Z

D

Z

D

χ_ǫ(z− y)K(z − y) dµ(y) dµ(z) = 0

because of the antisymmetry of K. Therefore, by (2.12), (2.13), (2.8), and the dominated convergence theorem, R

D

R_DcK(z− y) dµ(y)

 dµ(z) < ∞ (in particular, we have seen that E_m^aµ is well defined) and R

DT (χ_Dµ) dµ = 0. Using this and (2.12), we finally have that E_m^aµ(x) = 1

µ(D) Z

D

T (χ_D^cµ) dµ = 1 µ(D)

Z

D

T µ dµ (2.14)

for x∈ D ∈ Dm^a, thus E_m^aµ(x) is the average of the function T µ on the v-cube D∈ Dm^a which contains x. So, it is completely clear that, for a fixed a∈ Rⁿ,{Em^aµ}_m∈Zis a martingale. In [MV] it is shown that{Em^aµ}m∈Zis well defined and it is a martingale without the assumption of the existence of T µ (i.e., for more general measures µ).

Now, we can use (2.14), the L² boundedness of the dyadic maximal operator and (2.9) to deduce that

(2.15) kEm^aµkL²(µ) .kT µkL²(µ) .µ(P )^1/2

for all a∈ Rⁿand m∈ Z, where the constants that appear in the previous inequalities only depend on C₀, n, d and Lip(A).

Set E^aµ :={Em^aµ}m∈Z. Then, the martingale E^aµ belongs to L²(µ) by (2.15); thus by Theorem 2.4, for all a∈ Rⁿ,

kVρ(E^aµ)kL²(µ) .kE^aµkL²(µ) .µ(P )^1/2 for ρ > 2, kO(E^aµ)kL²(µ) .kE^aµkL²(µ) .µ(P )^1/2,

(2.16)

where the constants in the previous inequalities only depend on C₀, n, d, and Lip(A) (and on ρ, in the case ofVρ).

Finally, for x∈ R^d, we define

E_mµ(x) := 2^mn Z

{a : x∈Dm^a}

E_m^aµ(x) da

(notice that Lⁿ({a : x ∈ Dm^a}) = 2^−mn). Thus, E_mµ is an average (of the m’th term) of some martingales depending on a parameter a∈ Rⁿ.

Set Eµ := {Emµ}m∈Z. We want to obtain estimates like (2.16) for Vρ(Eµ) and O(Eµ).

We will only show the details for Vρ(Eµ), because the case of O(Eµ) follows by similar arguments.

One can easily check that E_mµ(x) = 2^{M n}R

[0,2^−M]ⁿE_m^aµ(x) da for all m, M ∈ Z with M ≤ m. Therefore, for all M, r, s ∈ Z with M ≤ r ≤ s, we have

(2.17) E_rµ(x)− Esµ(x) = 2^{M n} Z

[0,2^−M]ⁿ

(E_r^aµ(x)− Es^aµ(x)) da.

Given M ∈ Z, we consider the auxiliary transformation Vρ,M(Eµ)(x) := sup

{r^m}

 X

m∈Z

|Erm+1µ(x)− Ermµ(x)|^ρ

1/ρ

,

where the pointwise supremum is taken over all decreasing sequences of integers {rm}m∈Z

such that r_m ≥ M for all m ∈ Z. With this definition it is obvious that the sequence {Vρ,M(Eµ)(x)}M ∈Z is non increasing and Vρ(Eµ)(x) = lim_{M →−∞}Vρ,M(Eµ)(x) for all x ∈ R^d. Minkowski’s integral inequality and (2.17) yield the pointwise estimate

Vρ,M(Eµ)(x) = sup

{r^m} : r^m≥M

 X

m∈Z

|Erm+1µ(x)− Ermµ(x)|^ρ

1/ρ

≤ 2^{M n} Z

[0,2^−M]ⁿ

sup

{r^m}

 X

m∈Z

|Er^am+1µ(x)− Er^amµ(x))|^ρ

1/ρ

da

= 2^{M n} Z

[0,2^−M]ⁿVρ(E^aµ)(x) da.

Therefore, by the previous estimate, Minkowski’s integral inequality and (2.16), kVρ,M(Eµ)kL²(µ) ≤ 2^{M n}

Z

[0,2^−M]ⁿkVρ(E^aµ)kL²(µ)da≤ Cµ(P )^1/2,

where C > 0 only depends on C₀, n, d, Lip(A), and ρ. By the monotone convergence theorem, we conclude that kVρ(Eµ)kL²(µ) . µ(P )^1/2. Thus we have proved the following theorem (which can be considered the starting point to prove Theorem 1.1):

Theorem 2.7. Fix a cube eP ⊂ Rⁿ. Set Γ :={x ∈ R^d : x = (ex, A(ex))}, where A : Rⁿ → Rd−n is a Lipschitz function supported in eP , and set P := eP× R^d−n. Set µ := fH_Γⁿ, where f (x) = 1 for all ex∈ eP^c and C₀⁻¹≤ f(x) ≤ C0 for all ex∈ eP , for some constant C₀ > 0.

Let ρ > 2. Then, there exist constants C₁, C₂ > 0 such that kVρ(Eµ)kL²(µ) ≤ C1µ(P )^1/2 andkO(Eµ)kL²(µ) ≤ C2µ(P )^1/2, where C₁ and C₂ only depend on C₀, n, d, and Lip(A) (and on ρ in the case of C₁).

We need to introduce additional notation in order to express E_mµ in a more convenient way for our purposes. Let µ1, . . . , µ_kbe a finite collection of positive Borel measures such that µ_l(D_m^a) > 0 for all a∈ Rⁿ, m∈ Z and l = 1, . . . , k. Given m ∈ Z and x1, . . . , x_i, y₁, . . . , y_j ∈ R^d, we define

Λ^µ_m^1,...,µk(x₁, . . . , x_i; y₁, . . . , y_j) := 2^nm Z

{a : x1,...,xi∈Dm^a, y1,...,yj∈D/ m^a}

Qk da

l=1µ_l(D_m^a). Then, by Fubini’s theorem,

E_mµ(x) = Z

{a : x∈Dm^a}

2^mn µ(D_m^a)

Z

Dm^a

Z

(Dm^a)^c

K(z− y) dµ(y) dµ(z) da

= Z Z 

2^mn Z

{a : x,z∈Dm^a, y /∈Dm^a}

da µ(D_m^a)



K(z− y) dµ(z) dµ(y)

= Z Z

Λ^µ_m(x, z ; y)K(z− y) dµ(z) dµ(y).

(2.18)

3. Sketch of the proof of Theorem 1.1

The proof relies on two basic facts: the known L² boundedness of the ρ-variation and oscillation of martingales explained in the previous section and the good geometric properties of Lipschitz graphs from a measure-theoretic point of view.

As we said above, the starting point of the proof is Theorem 2.7, where the L²boundedness of the ρ-variation and oscillation (of a convex combination) of some particular martingales is stated. So, the first step consists in relating the results on martingales in Theorem 2.7 with the ρ-variation and oscillation of singular integrals on Lipschitz graphs, and this is the aim of the following two theorems:

Theorem 3.1. Let Γ and µ be as in Theorem 2.7. For each x∈ Γ, define

(3.1) W µ(x)² := X

m∈Z

|(Kϕ2^−m∗ µ)(x) − Emµ(x)|². Then, kW µk²_L²_(µ) ≤ C1P

Q∈D α_µ(C₂Q)²+ β_2,µ(Q)²

µ(Q), where C₁, C₂ > 0 depend only on C₀, n, d, K, and Lip(A).

Theorem 3.2. Let Γ and µ be as in Theorem 2.7. For each x∈ Γ, define

(3.2) Sµ(x)²:= sup

{ǫ^m}

X

j∈Z

X

m∈Z: ǫ^m,ǫm+1∈I^j

|(Kϕǫ^ǫm+1^m ∗ µ)(x)|²,

where I_j = [2^−j−1, 2^−j) and the supremum is taken over all decreasing sequences of positive numbers {ǫm}m∈Z. Then, kSµk²L²(µ) ≤ CP

Q∈D α_µ(Q)²+ β_2,µ(Q)²

µ(Q), where C > 0 only depends on C0, n, d, K, and Lip(A).

Two fundamental tools to study W µ and Sµ are the α and β coefficients, which will be used to measure the flatness of Γ at different scales, in order to estimate the terms which appear in the sums in (3.1) and (3.2). This will be done in sections 4 and 5. To use the α coefficients to relate the ρ-variation of martingales with the ρ-variation of singular integrals, it is a key fact that we are considering a “smooth” family like ϕ, because the α’s are defined in terms of Lipschitz functions but T_ǫ is defined by means of a rough truncation. Moreover, we are taking a truncation only on the first n-coordinates because the average of martingales that we are using is taken over the parameter a∈ Rⁿ, using the v-cubes D_M^a (see subsection 2.3).

Combining Theorem 3.1 and Theorem 3.2 with the L² estimates of the ρ-variation and oscillation on the average of martingales Eµ in Theorem 2.7, we are able to obtain local L² estimates of Vρ◦ Tϕ^HnΓ and O ◦ Tϕ^HnΓ when Γ is any Lipschitz graph. More precisely, we separate the sum in the definition of Vρ◦ Tϕ^HnΓ into two parts, which are classically called short and long variation (and analogously forO ◦ T^ϕHnΓ). The short variation corresponds to the sum Sµ in Theorem 3.2 (here µ is a suitable modification ofHⁿ_Γ), where the indices run over m∈ Z such that both ǫm and ǫ_m+1 lie in the same dyadic interval, and can be handled using the α’s and β’s. The long variation corresponds to the sum over the indices m∈ Z such that ǫ_m and ǫ_m+1 lie in different dyadic intervals, so one may assume that the ǫ_m’s are dyadic numbers. It is handled by comparing Kϕ₂^−m∗ µ with Emµ, and then using Theorem 3.1 and the fact the ρ-variation and oscillation of Eµ are bounded in L²(µ), by Theorem 2.7.

This will be done in section 6 (see Theorem 6.1).

Using the local L² estimates of Theorem 6.1, combined with rather standard techniques in Calder´on-Zygmund theory, in section 7 we obtain the H¹(HΓⁿ)→ L¹(HΓⁿ) and L^∞(HΓⁿ)→ BM O(HⁿΓ) boundedness of Vρ◦ Tϕ^HnΓ and O ◦ Tϕ^HnΓ. Then, by interpolation, we obtain the L^p boundedness of these operators in the whole range 1 < p <∞, and in particular the L² boundedness (see Theorem 7.1). Moreover, [CJRW2, Theorem B] can be adapted to prove that the L²(HⁿΓ) boundedness ofVρ◦ T^ϕHnΓ andO ◦ T^ϕHnΓ also yields the boundedness of these operators from L¹(HⁿΓ) to L^1,∞(HⁿΓ).

Let us stress that almost all the estimates in the proof of Theorem 1.1 (in particular, the constants involved in the relationships ., & and≈) depend either on n, d, K or Lip(A), and possibly on other variables such as ρ or p.

4. Proof of Theorem 3.1

In order to study the difference (Kϕ₂^−m∗µ)(x)−Emµ(x), we are going to split E_mµ(x) into two parts, the one we will compare with (Kϕ₂^−m∗ µ)(x) (which corresponds to integrate, in the definition of E_mµ(x), over the points y∈ R^dsuch that 2^−m .|ex− ey|), and the remaining part. Then, we will estimate each part of (Kϕ₂^−m ∗ µ)(x) − Emµ(x) separately, using the cancelation properties of the kernel K and the uniform rectifiability of µ.

Recall from (2.18) that E_mµ(x) = RR

Λ^µ_m(x, z ; y)K(z− y) dµ(z) dµ(y). Given ǫ > 0, we set γ_ǫ:= 1− ϕǫ. Then,

E_mµ(x) = Z Z

ϕ₂^−m(x− y)Λ^µm(x, z ; y)K(z− y) dµ(z) dµ(y) +

Z Z

γ₂^−m(x− y)Λ^µm(x, z ; y)K(z− y) dµ(z) dµ(y).

The first term in the previous sum is the one that we will compare with (Kϕ₂^−m∗µ)(x). For all a∈ Rⁿsuch that x∈ Dm^a, we have supp ϕ₂^−m(x−·)∩Dm^a =∅, and thus (Kϕ2^−m∗µ)(x) = (Kϕ₂^−m∗ (χ(D_m^a)^cµ))(x). Hence, using Fubini’s theorem and the definition of Λ^µm(x, z ; y),

(Kϕ₂^−m∗ µ)(x) = 2^mn Z

{a : x∈Dm^a}

(Kϕ₂^−m∗ (χ(D_m^a)^cµ))(x) da

= 2^mn Z

{a : x∈Dm^a}

µ(D_m^a)⁻¹ Z

D_m^a

(Kϕ₂^−m∗ (χ(Dm^a)^cµ))(x) dµ(z) da

= Z Z

ϕ₂^−m(x− y)Λ^µm(x, z ; y)K(x− y) dµ(z) dµ(y).

We can decompose (Kϕ₂^−m∗ µ)(x) − Emµ(x) as (Kϕ₂^−m∗ µ)(x) − Emµ(x)

= ZZ

ϕ₂^−m(x− y)Λ^µm(x, z ; y)(K(x− y) − K(z − y)) dµ(z) dµ(y)

− ZZ

γ₂^−m(x− y)Λ^µm(x, z ; y)K(z− y) dµ(z) dµ(y)

= X

j<m

F_j^m(x)−X

j∈Z

G^m_j (x), (4.1)

where

F_j^m(x) :=

ZZ

ϕ₂^2−−jj−1(x− y)Λ^µm(x, z ; y)(K(x− y) − K(z − y)) dµ(z) dµ(y), (4.2)

G^m_j (x) :=

ZZ

ϕ₂²⁻⁻j−1^j (z− y)γ2^−m(x− y)Λ^µm(x, z ; y)K(z− y) dµ(z) dµ(y).

(4.3)

Fix a v-cube D ∈ Dm, for some m∈ Z. In subsection 4.1 (see (4.18)) we will prove that

(4.4) X

j<m

|Fj^m(x)| . dist(x, L_D)

ℓ(D) + X

Q∈D : D⊂Q

ℓ(D) ℓ(Q)α(Q)